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A239621 Gaussian primes x + i*y, with x = a(2n-1) >= y = a(2n) >= 0, sorted by norm.

%I A239621 #38 Feb 16 2025 08:33:21
%S A239621 1,1,2,1,3,0,3,2,4,1,5,2,6,1,5,4,7,0,7,2,6,5,8,3,8,5,9,4,10,1,10,3,8,
%T A239621 7,11,0,11,4,10,7,11,6,13,2,10,9,12,7,14,1,15,2,13,8,15,4,16,1,13,10,
%U A239621 14,9,16,5,17,2,13,12,14,11,16,9,18,5,17,8,19,0
%N A239621 Gaussian primes x + i*y, with x = a(2n-1) >= y = a(2n) >= 0, sorted by norm.
%C A239621 The condition a >= b >= 0 implies that there is only one Gaussian prime for each norm. - _T. D. Noe_, Mar 26 2014
%C A239621 The real parts and imaginary parts are listed as a(2n-1) = A300587(n) and a(2n) = A300588(n), respectively. Sequence A239397 lists the pair (y, x) after each pair (x, y), except for (1, 1). - _M. F. Hasler_, Mar 10 2018
%H A239621 T. D. Noe, <a href="/A239621/b239621.txt">Table of n, a(n) for n = 1..10106</a> (5053 pairs)
%H A239621 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GaussianPrime.html">Gaussian prime</a>
%H A239621 Wikipedia, <a href="http://en.wikipedia.org/wiki/Complex_number">Complex Number</a>
%e A239621 From _M. F. Hasler_, Mar 09 2018: (Start)
%e A239621 Sorted by norm, the smallest Gaussian primes z = x + iy in the first half-quadrant x >= y >= 0 are:
%e A239621 a(1) + i*a(2) = 1 + i;
%e A239621 a(3) + i*a(4) = 2 + i;
%e A239621 a(5) + i*a(6) = 3;
%e A239621 ... (End)
%t A239621 mx = 20; lst = Flatten[Table[{a, b}, {a, 0, mx}, {b, 0, a}], 1]; qq = Select[lst, Norm[#] <= mx && PrimeQ[#[[1]] + I*#[[2]], GaussianIntegers -> True] &]; Sort[qq, Norm[#1] < Norm[#2] &]
%o A239621 (PARI) {for(n=2,400, f=factor(n*I)/*factor in Z[i]*/; matsize(f)[1]<=2 && vecsum(f[,2])==2+(f[1,1]==I) /*either I*p^2 or w*conj(w/I), maybe (1+I)^2 */ && printf("%d,",vecsort([real(f=f[3-f[1,2],1]),imag(f)],,4)))} \\ For illustrative use. - _M. F. Hasler_, Mar 09 2018
%Y A239621 Cf. A055025 (norms of Gaussian primes), A239397.
%Y A239621 Cf. A300587, A300588.
%K A239621 nonn
%O A239621 1,3
%A A239621 _T. D. Noe_, Mar 22 2014
%E A239621 Name changed and in cf. complex -> Gaussian - _Wolfdieter Lang_, Mar 25 2014
%E A239621 Name edited by _M. F. Hasler_, Mar 09 2018